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9.3
Graphing Quadratic Functions
What you should learn
GOAL 1 Sketch the graph
of a quadratic function.
GOAL 1
SKETCHING A QUADRATIC FUNCTION
A quadratic function is a function that can be written in the standard form
y = ax 2 + bx + c, where a ≠ 0.
Use quadratic
models in real-life settings,
such as finding the winning
distance of a shot put in
Example 3.
Every quadratic function has a U-shaped graph called a parabola. If the leading
coefficient a is positive, the parabola opens up. If the leading coefficient is
negative, the parabola opens down in the shape of an upside down U.
Why you should learn it
The graph on the left has a positive leading coefficient so it opens up. The graph
on the right has a negative leading coefficient so it opens down.
GOAL 2
RE
y
y
(3, 9)
y x2
FE
To model real-life
parabolic situations, such as
the height above the water of
a jumping dolphin in
Exs. 65 and 66.
AL LI
(2, 4)
9
5
(3, 9)
(1, 3)
3
y x 2 4
7
5
(2, 0)
(2, 4)
5
3
1
5
(1, 1)
3
(0, 0)
3
1
1
1
(2, 0)
1
3
5 x
3
(1, 1)
1
(0, 4)
(1, 3)
3
(3, 5)
5 x
5
(3, 5)
The vertex is the lowest point of a parabola that opens up and the highest point
of a parabola that opens down. The parabola on the left has a vertex of (0, 0) and
the parabola on the right has a vertex of (0, 4).
The line passing through the vertex that divides the parabola into two symmetric
parts is called the axis of symmetry. The two symmetric parts are mirror
images of each other.
G R A P H O F A Q UA D R AT I C F U N C T I O N
The graph of y = ax 2 + bx + c is a parabola.
518
•
If a is positive, the parabola opens up.
•
•
•
If a is negative, the parabola opens down.
b
2a
The vertex has an x-coordinate of º.
b
2a
The axis of symmetry is the vertical line x = º.
Chapter 9 Quadratic Equations and Functions
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G R A P H I N G A Q UA D R AT I C F U N C T I O N
STEP 1
Find the x-coordinate of the vertex.
STEP 2
Make a table of values, using x-values to the left and right of
the vertex.
STEP 3
Plot the points and connect them with a smooth curve to form
a parabola.
EXAMPLE 1
Graphing a Quadratic Function with a Positive a-value
Sketch the graph of y = x 2 º 2x º 3.
SOLUTION
1
Find the x-coordinate of the vertex when a = 1 and b = º2.
b
2a
º2
2(1)
º = º = 1
2
(2, 5)
Make a table of values, using x-values to the
left and right of x = 1.
x
º2
5
y
3
y
º1
0
0
º3
1
2
º4
3
º3
0
4
5
(4, 5)
y x 2 2x 3
1
(1, 0)
(3, 0)
3
Plot the points. The vertex is (1, º4) and the
axis of symmetry is x = 1. Connect the points
to form a parabola that opens up since a is positive.
EXAMPLE 2
5
4 x
1
(2, 3)
(1, 4)
(0, 3)
Graphing a Quadratic Function with a Negative a-value
Sketch the graph of y = º2x 2 º x + 2.
SOLUTION
1
Find the x-coordinate of the vertex when a = º2 and b = º1.
b
2a
º1
2(º2)
1
4
º = º = º
STUDENT HELP
2
Study Tip
If the x-coordinate of the
vertex is a fraction, you
can still choose whole
numbers when you make
a table.
3
1
4
Make a table of values, using x-values to the left and right of x = º.
x
º2
º1
y
º4
1
1
4
1
2 8
º
0
1
2
2
º1
º8
1 1
Plot the points. The vertex is º, 2 and the
4 8
1
axis of symmetry is x = º . Connect the
4
points to form a parabola that opens down
since a is negative.
14, 2 18
(1, 1)
3
(2, 4)
y
(0, 2)
1
1
3
x
(1, 1)
3
y 2x 2 x 2
5
9.3 Graphing Quadratic Functions
519
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GOAL 2
USING QUADRATIC MODELS IN REAL LIFE
When an object has little air resistance, its path through the air can be
approximated by a parabola.
FOCUS ON
PEOPLE
EXAMPLE 3
Using a Quadratic Model
TRACK AND FIELD Natalya Lisovskaya holds the world record for the
women’s shot put. The path of her record-breaking throw can be modeled by
y = º0.01347x 2 + 0.9325x + 5.5, where x is the horizontal distance in feet and
y is the height (in feet). The initial height is represented by 5.5, the height at
which the shot (a 4-kilogram metal ball) was released.
a. What was the maximum height (in feet) of the shot thrown by Lisovskaya?
b. What was the distance of the throw to the nearest hundredth of a foot?
SOLUTION
a. The maximum height of the throw occurred at the vertex of the parabolic
RE
FE
L
AL I
path. Find the x-coordinate of the vertex. Use a = º0.01347 and b = 0.9325.
NATALYA
LISOVSKAYA
In 1987, the women’s world
record in shot put was set by
Lisovskaya. She also won an
Olympic gold medal in 1988.
b
2a
0.9325
2(º0.01347)
º = º ≈ 34.61
Substitute 34.61 for x in the model to find the maximum height.
y = º0.01347(34.61)2 + 0.9325(34.61) + 5.5 ≈ 21.6
The maximum height of the shot was about 21.6 feet.
b. To find the distance of the throw, sketch the parabolic path of the shot.
Because a is negative, the graph opens down. Use (35, 22) as the vertex. The
y-intercept is 5.5. Sketch a symmetric curve.
Height (ft)
y
30
y 0.01347x 2 0.9325x 5.5
maximum height
20
(69.2, 5.5)
10
0
maximum
distance
(0, 5.5)
0
10
20
30
40
50
60
70
80
90 x
Horizontal distance (ft)
The distance of the throw is the x-value that yields a y-value of 0. The graph
shows that the distance was between 74 and 75 feet. You can use a table to
refine this estimate.
Distance, x
Height, y
520
74.5 ft
74.6 ft
74.7 ft
74.8 ft
0.209 ft
0.102 ft
º0.006 ft
º0.114 ft
The shot hit the ground at a distance between 74.6 and 74.7 feet.
Chapter 9 Quadratic Equations and Functions
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GUIDED PRACTICE
Vocabulary Check
✓
1. Identify the values of a, b, and c for the quadratic function in standard form
y = º5x 2 + 7x º 4.
2. Why is the vertical line that passes through the vertex of a parabola called the
Concept Check
✓
axis of symmetry?
3. Explain how you can decide whether the graph of y = 3x 2 + 2x º 4 opens
up or down.
4. Find the coordinates of the vertex of the graph of y = 2x 2 + 4x º 2.
Skill Check
✓
Tell whether the graph opens up or down. Write an equation of the axis of
symmetry.
5. y = x 2 + 4x º 1
6. y = 3x 2 + 8x º 6
8. y = ºx 2 º 4x + 2
9. y = 5x 2 º 2x + 4
7. y = x 2 + 7x º 1
10. y = ºx 2 + 4
Sketch the graph of the function. Label the vertex.
11. y = º3x 2
12. y = º3x 2 + 6x + 2
13. y = º5x 2 + 10
14. y = x 2 + 4x + 7
15. y = x 2 º 6x + 8
16. y = 5x 2 + 5x º 2
17. y = º4x 2 º 4x + 12
18. y = 3x 2 º 6x + 1
19. y = 2x 2 º 8x + 3
20.
BASKETBALL You throw a basketball whose path can be modeled by
y = º16x 2 + 15x + 6, where x represents time (in seconds) and y represents
height of the basketball (in feet).
a. What is the maximum height that the basketball reaches?
b. In how many seconds will the basketball hit the ground if no one catches it?
PRACTICE AND APPLICATIONS
STUDENT HELP
PREPARING TO GRAPH Complete these steps for the function.
Extra Practice
to help you master
skills is on p. 805.
a. Tell whether the graph of the function opens up or down.
b. Find the coordinates of the vertex.
c. Write an equation of the axis of symmetry.
21. y = 2x 2
22. y = º7x 2
23. y = 6x 2
1
24. y = x 2
2
25. y = º5x 2
26. y = º4x 2
27. y = º16x 2
28. y = 5x 2 º x
29. y = 2x 2 º 10x
30. y = º7x 2 + 2x
31. y = º10x 2 + 12x
STUDENT HELP
32. y = 6x 2 + 2x + 4
33. y = 5x2 + 10x + 7
34. y = º4x 2 º 4x + 8
HOMEWORK HELP
35. y = 2x 2 º 7x º 8
36. y = 2x 2 + 7x º 21
37. y = ºx 2 + 8x + 32
Example 1: Exs. 21–64
Example 2: Exs. 21–64
Example 3: Exs. 65–75
1
38. y = x 2 + 3x º 7
2
1
39. y = 4x 2 + x º 8
4
40. y = º10x 2 + 5x º 3
41. y = 0.78x 2 º 4x º 8
42. y = 3.5x 2 + 2x º 8
43. y = º10x 2 º 7x + 2.66
9.3 Graphing Quadratic Functions
521
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SKETCHING GRAPHS Sketch the graph of the function. Label the vertex.
44. y = x 2
45. y = º2x 2
46. y = 4x 2
47. y = x 2 + 4x º 1
48. y = º3x 2 + 6x º 9
49. y = 4x 2 + 8x º 3
50. y = 2x 2 º x
51. y = 6x 2 º 4x
52. y = 3x 2 º 2x
53. y = x 2 + x + 4
55. y = 3x 2 º 2x – 1
56. y = 2x 2 + 6x º 5
1
54. y = x 2 + x + 4
57. y = º3x 2 º 2x º 1
59. y = º4x 2 + 4x + 7
60. y = º3x 2 º 3x + 4
61. y = º2x 2 + 6x º 5
1
62. y = ºx 2 + 2x º 3
3
1
63. y = ºx 2 º 4x + 6
2
1
64. y = ºx 2 º x º 1
4
DOLPHIN In Exercises 65 and 66,
use the following information.
h
6
Height (ft)
A bottlenose dolphin jumps out of the
water. The path the dolphin travels can
be modeled by h = º0.2d 2 + 2d, where
h represents the height of the dolphin
and d represents horizontal distance.
58. y = º4x 2 + 32x º 20
4
2
0
65. What is the maximum height the
0
2
dolphin reaches?
4
6
8
10
d
Distance (ft)
66. How far did the dolphin jump?
FOCUS ON
APPLICATIONS
WATER ARC In Exercises 67 and 68, use the following information.
On one of the banks of the Chicago River, there is a water cannon, called the
Water Arc, that sprays recirculated water across the river. The path of the Water
Arc is given by the model
y = º0.006x 2 + 1.2x + 10
where x is the distance (in feet) across the river, y is the height of the arc (in feet),
and 10 is the number of feet the cannon is above the river.
67. What is the maximum height of the water sprayed from the Water Arc?
68. How far across the river does the water land?
GOLD PRODUCTION In Exercises 69–71, use the following information.
WATER ARC
To celebrate the
engineering feat of reversing
the flow of the Chicago River,
the Water Arc was built on
the hundredth anniversary of
this event.
which years was the
production of gold in Ghana
decreasing?
70. From 1980 to 1995, during
which years was the
production of gold increasing?
71. How are the questions asked in
Exercises 69 and 70 related to
the vertex of the graph?
522
Gold Production in Ghana
69. From 1980 to 1995, during
Chapter 9 Quadratic Equations and Functions
G
Ounces (thousands)
RE
FE
L
AL I
In Ghana from 1980 to 1995, the annual production of gold G in thousands of
ounces can be modeled by G = 12t 2 º 103t + 434, where t is the number of
years since 1980.
1500
1000
500
0
0
4
16
8
12
Years since 1980
t
Page 6 of 7
TABLE TENNIS In Exercises 72–75, use the following information.
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 72–75.
Suppose a table-tennis ball is hit in such a way that its path can be modeled
by h = º4.9t 2 + 2.07t, where h is the height in meters above the table and t is
the time in seconds.
72. Estimate the maximum height
reached by the table-tennis ball.
Round to the nearest tenth.
73. About how many seconds did it
take for the table-tennis ball to
reach its maximum height after
its initial bounce? Round to the
nearest tenth.
74. About how many seconds did it take for the table-tennis ball to travel from
the initial bounce to land on the other side of the net? Round to the nearest
tenth.
75. CRITICAL THINKING What factors would change the path of the table-tennis
ball? What combination of factors would result in the table-tennis ball
bouncing the highest? What combination of factors would result in the tabletennis ball bouncing the lowest?
Test
Preparation
76.
MULTI-STEP PROBLEM A sprinkler can eject water at an angle of 35°,
60°, or 75° with the ground. For these settings, the paths of the water can be
modeled by the equations below where x and y are measured in feet.
35°: y = º0.06x 2 + 0.70x + 0.5
y
60°: y = º0.16x 2 + 1.73x + 0.5
C
6
B
75°: y = º0.60x 2 + 3.73x + 0.5
14
10
6
2
A
2
6
10
14 x
a. Find the maximum height of the water for each setting.
b. Find how far from the sprinkler the water reaches for each setting.
c. CRITICAL THINKING Do you think there is an angle setting for the
sprinkler that will reach farther than any of the settings above? How do the
angle and reach represented by the graph of y = º0.08x 2 + x + 0.5
compare with the others? What angle setting would reach the least distance?
★ Challenge
77. UNDERSTANDING GRAPHS Sketch the graphs of the three functions in the
same coordinate plane. Describe how the three graphs are related.
a. y = x 2 + x + 1
1
y = x 2 + x + 1
2
y = 2x 2 + x + 1
b. y = x 2 º 1x + 1
c. y = x 2 º x + 1
y = x 2 º 5x + 1
y = x2 º x + 3
y = x 2 º 10x + 1
y = x2 º x º 2
78. How does a change in the value of a change the graph of y = ax 2 + bx + c?
EXTRA CHALLENGE
www.mcdougallittell.com
79. How does a change in the value of b change the graph of y = ax 2 + bx + c?
80. How does a change in the value of c change the graph of y = ax 2 + bx + c?
9.3 Graphing Quadratic Functions
523
Page 7 of 7
MIXED REVIEW
GRAPHING Write the equation in slope-intercept form, and then graph the
equation. Label the x- and y-intercepts on the graph. (Review 4.6 for 9.4)
81. º3x + y + 6 = 0
82. ºx + y º 7 = 0
83. 4x + 2y º 12 = 0
84. x + 2y º 7 = 5x + 1
GRAPHING LINEAR INEQUALITIES Graph the system of linear inequalities.
(Review 7.6)
85. x º 3y ≥ 3
86. x + y ≤ 5
x º 3y ≤ 12
87. x + y < 10
x≥2
y≥0
2x + y > 10
xºy<2
SIMPLIFYING EXPRESSIONS Simplify. Write your answer as a power or as
an expression containing powers. (Review 8.1)
88. 45 • 48
89. (33)2
90. (36)3
91. a • a5
92. (3b4)2
93. 6x • (6x)2
94. (3t)3(ºt 4)
95. (º3a2b2)3
SCIENTIFIC NOTATION Rewrite the number in scientific notation.
(Review 8.4)
96. 0.0012
97. 987,000
99. 1,229,000,000
98. 3,984,328
100. 0.000432
101. 0.00999
QUIZ 1
Self-Test for Lessons 9.1–9.3
Evaluate the expression. Give the exact value if possible. Otherwise,
approximate to the nearest hundredth. (Review 9.1)
1. 1
44
2. º1
96
3. º6
76
4. º2
7
5. 6
6. 1
.5
7. 0
.1
6
8. 2
.2
5
Solve the equation. (Review 9.1)
9. x 2 = 169
10. 4x 2 = 64
11. 12x 2 = 120
12. º6x 2 = º48
21
2
1
15. 4
16.
Simplify the expression. (Review 9.2)
13. 1
8
14. 5
• 20
4356
Tell whether the graph of the function opens up or down. Find the
coordinates of the vertex. Write the equation of the axis of symmetry
of the function. (Review 9.3)
17. y = x 2 + 2x º 11
18. y = 2x 2 º 8x º 6
19. y = 3x 2 + 6x º 10
1
20. y = x 2 + 5x º 3
2
21. y = 7x 2 º 7x + 7
22. y = x 2 + 9x
Sketch a graph of the function. (Review 9.3)
23. y = ºx 2 + 5x º 5
524
24. y = 3x 2 + 3x + 1
Chapter 9 Quadratic Equations and Functions
25. y = º2x 2 + x º 3